In these expressions, \(\rho _i({\bf r})\)  (\({\bf r}=r\hat{{\bf r}})\)  is the three-dimensional distribution function around the atom \(i\) , \(\rho _i({\bf r})=\sum _{j\ne i}^{N_{\textup {a}}}\tilde{\rho }_{ij}({\bf r})\) , and \(\tilde{\rho }_{ij}({\bf r})\)  is the likelihood to find
atom \(j\)  at position \({\bf r}\)  from atom \(i\)  within a certain cutoff radius.
Essentially, \(\rho _i^{(2)}(r)\)  yields the probability to find an atom \(j\)  (\(j\ne i\) )
at a distance \(r\)  from atom \(i\) , and \(\rho _i^{(3)}(r,s,\theta )\)  the probability
to find two atoms \(j,k\)  at a distance \(r,s\) , respectively, from atom \(i\)  (\(j\ne i\) ,
\(k\ne j,i\) ) and spanning the angle \(\theta \)  between them.
In practice, these distribution functions are expressed as a combination of
radial and angular basis functions, with the coefficients forming a set of
descriptors that we simply denote as \(\mathbf {x}_i^{(2)}\)  and
\(\mathbf {x}_i^{(3)}\) . Following [1]}, we define \(\mathbf {x}_i=(\mathbf {x}_i^{(2)},\mathbf {x}_i^{(3)})\) .
